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Gudermann's idea was to base everything on the power series
expansion of the functions. (This statement will have to do for
the moment; its meaning mil become clear when we describe
one of the leading motivations of the work of Weierstrass.) This
really was a good new idea, and Gudermann slaved over it with
overwhelming German thoroughness for years without, perhaps,
realizing what lay behind his inspiration, and himself never
carried it through. The important thing to note here is that
Weierstrass made the theory of power series - Gudermann's
inspiration - the nerve of all his work in analysis. He got the
idea from Gudermann, whose lectures he attended. In later life,
contemplating the scope of the methods he had developed in
analysis, Weierstrass was wont to exclaim, "There is nothing
but power series!*
At the opening lecture of Gudermann's course on elliptic
functions (he called them by a different name, but that is of no
importance) there were thirteen auditors. Being in love with his
subject the lecturer quickly left the earth and was presently
soaring practically alone in the aether of pure thought. At the
second lecture only one auditor appeared and Gudermann was
happy. The solitary student was Weierstrass. Thereafter no
incautious third party ventured to profane the holy communion
between the lecturer and his unique disciple. Gudermann and
Weierstrass were fellow Catholics; they got along splendidly
Weierstrass was duly grateful for the pains Gudermann
lavished on him, and after he had become famous he seized
every opportunity - the more public the better - to proclaim
his gratitude for what Gudermann had done for him. The debt
was not inconsiderable: it is not every professor who can drop
a faint like the one - power series representation of functions as
a point of attack - which inspired Weierstrass. In addition to
the lectures on elliptic functions, Gudermann also gave Weier-
strass private lessons on 'analytical spherics* - whatever that
may have been.
In 1841, at the age of twenty-six, Weierstrass took his exam-
inations for his teacher's certificate. The examination was in
two sections, written and oral. For the first he was allowed six